From Putty:
Login to putty(assuming that you using linux)
type following command
mysqldump -u root -pPASSWORDHERE database > filename.sql
it will create filename.sql in the folder you are in, download that filename.sql to your pc, connect to your new vps, transfer filename.sql to there.
then open putty and type following command
mysql -u root -pPASSWORDHERE database < filename.sql
OR
If using PhpMyAdmin:
select a database, and export the database by button located at top, choose appropriate options, and export it,(file will be downloaded.)
On New Host:
Login to phpmyadmin,create database, select that database, Press Import button located at top(besides Export Button), Choose your sql file and submit, and your database will be transferred.